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成功大學 111 年度 微積分C

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第 1 題10 分
(a)5 分
Evaluate the limit lim⁡n→∞3n+5nn\lim\limits_{n \to \infty} \sqrt[n]{3^n + 5^n}.
(b)5 分
Find the horizontal asymptote of the graph y=x2(21x−21x2)y = x^2(2^{\frac{1}{x}} - 2^{\frac{1}{x^2}}) for x>0x > 0 if it exists.
第 2 題10 分
Define f(x)=∫2xx21+sin⁡(πt)dtx−2f(x) = \frac{\int_{2x}^{x^2} \sqrt{1 + \sin(\pi t)} dt}{x - 2} for x≠2x \neq 2. Give a value of f(2)f(2) such that ff is continuous at 2.
第 3 題10 分
Let f(x)=13x3+x+1f(x) = \frac{1}{3}x^3 + x + 1 and g=f−1g = f^{-1} be the inverse function of ff. A curve CC satisfies the equation 2x2y+xy2=82x^2y + xy^2 = 8. Find a point (a,b)(a, b) such that (i) (a,b)(a, b) is in the first quadrant, (ii) (a,b)(a, b) is on the graph of gg, and (iii) the tangent line to the graph of gg at (a,b)(a, b) is perpendicular to the tangent line to the curve CC at (1,2)(1, 2)
第 4 題10 分
Let f(x)=ln⁡(1−x−2x2)f(x) = \ln(1 - x - 2x^2).
(a)5 分
Find the Taylor expansion for ff about x=0x = 0. (In the form ∑k=0∞akxk\sum\limits_{k=0}^{\infty} a_k x^k with a general formula for aka_k)
(b)5 分
Find the radius of convergence of the Taylor expansion in Problem(a).
第 5 題10 分
Let u(x,y)u(x, y) be a differentiable function with ∂u∂x(4,1)=1\frac{\partial u}{\partial x}(4, 1) = 1 and ∂u∂y(4,1)=2\frac{\partial u}{\partial y}(4, 1) = 2. Suppose that x=stx = st and y=sty = \frac{s}{t} and define h(s,t)=u(x(s,t),y(s,t))h(s, t) = u(x(s, t), y(s, t)). At the point (2,2)(2, 2), find a unit vector u\mathbf{u} in the stst-plane such that hh increases most rapidly in the direction.
第 6 題10 分
Suppose that f(π)=4f(\pi) = 4 and ∫0π[f(x)+f′′(x)]sin⁡xdx=5\int_0^\pi [f(x) + f''(x)] \sin x dx = 5. Find f(0)f(0).
第 7 題10 分
Find the arc length of the part of the curve r=1+sin⁡θr = 1 + \sin \theta which is inside the curve r=3sin⁡θr = 3 \sin \theta (the solid curve in the figure).
第 7 題圖表
第 8 題10 分
Find the maximum of f(x,y,z)=2x+7y−3zf(x, y, z) = 2x + 7y - 3z on the ellipsoid 2x2+7y2+3z2=62x^2 + 7y^2 + 3z^2 = 6.
第 9 題10 分
Evaluate the double integral
∬De2x−y2x+ydA\iint_D e^{\frac{2x-y}{2x+y}} dA
where DD is the trapezoid in the first quadrant with vertices (2,0)(2, 0), (4,0)(4, 0), (0,4)(0, 4) and (0,8)(0, 8).
第 10 題10 分
Let F(x,y,z)=xx2+y2+z2i+yx2+y2+z2j+zx2+y2+z2k\mathbf{F}(x, y, z) = \frac{x}{x^2 + y^2 + z^2}\mathbf{i} + \frac{y}{x^2 + y^2 + z^2}\mathbf{j} + \frac{z}{x^2 + y^2 + z^2}\mathbf{k}. Compute the surface integral
∬SF⋅dS\iint_S \mathbf{F} \cdot d\mathbf{S}
(using the outward pointing normal), when SS is the surface x2+y2+z2=225x^2 + y^2 + z^2 = 225.
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